Positive semidefinite diagonal minus tail forms are sums of squares

By a diagonal minus tail form (of even degree) we understand a real homogeneous polynomial F(x1, ..., xn) = F(x) = D(x) − T(x), where the diagonal part D(x) is a sum of terms of the form bix2d i with all bi ≥ 0 and the tail T(x) a sum of terms ai1i2...inxi1 1 ...xin n with ai1i2...in > 0 and at least two i ≥ 1. We show that an arbitrary change of the signs of the tail terms of a positive semidefinite diagonal minus tail form will result in a sum of squares of polynomials

The Hankel Pencil Conjecture

The Toeplitz pencil conjecture stated in [SS1] and [SS2] is equivalent to a conjecture for n £ n Hankel pencils of the form Hn(x) = (ci+j¡n+1); where c0 = x is an indeterminate, cl = 0 for l < 0; and cl 2 C¤ = Cn f0g; for l ¸ 1: In this paper it is shown to be implied by another conjecture, we call root conjecture. This latter claims for a certain pair (mnn;mn¡1;n) of submaximal minors of certain special Hn(x) that, viewed as elements of C[x]; there holds that roots(mnn) µ roots(mn¡1;n) implies roots(mn¡1;n) = f1g: We give explicit formulae in the ci for these minors and show the root conjecture for minors mnn;mn¡1;n of degree · 6: This implies the Hankel Pencil conjecture for matrices up to size 8 £ 8: Main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous, and use of the Sylvester identity for matrices

The Hankel pencil conjecture

AbstractThe Toeplitz pencil conjecture stated in [W. Schmale, P.K. Sharma, Problem 30-3: singularity of a toeplitz matrix, IMAGE 30 (2003); W. Schmale, P.K. Sharma, Cyclizable matrix pairs over C[x] and a conjecture on toeplitz pencils, Linear Algebra Appl. 389 (2004) 33-42] is equivalent to a conjecture for n×n Hankel pencils of the form Hn(x)=(ci+j-n+1), where c0=x is an indeterminate, cl=0 for l<0, and cl∈C∗=C⧹{0}, for l⩾1. In this paper it is shown to be implied by another conjecture, which we call the root conjecture. The root conjecture asserts a strong relationship between the roots of certain submaximal minors of Hn(x) specialized to have c1=c2=1. We give explicit formulae in the ci for these minors and prove the root conjecture for minors mnn,mn-1,n of degree ⩽6. This implies the Hankel Pencil conjecture for matrices up to size 8×8. The main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous, and use of the Sylvester identity for matrices

Quadrados Mágicos

[Excerto do Artigo]Mago, Magia, Mágico: Derivando das palavras gregas mágos, mageía,magikós (μαγoσ, μαγηια, μαγικoσ), enciclopédias e dicionários tradicionais e digitais (...) associam-nas invariavelmente a feiticeiros, astrólogos, sacerdotes da religião de Zoroastro; às ciência e arte que pretendem actuar sobre a natureza para aparentemente contrariar as suas leis; a encanto, fascínio e sedução, respectivamente. Para um matemático moderno, mesmo que nunca antes tivesse pensado no assunto, a existência de quadrados mágicos de todas as ordens, e ainda em grande quantidade, não surpreende..

A Note on Extrema of Linear Combinations of Elementary Symmetric Functions

This note provides a new approach to a result of Foregger and related earlier results by Keilson and Eberlein. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof given by Foregger is flawed.Comment: (v2) revision based on suggestions by refere

On the corners of certain determinantal ranges

Let A be a complex n×n matrix and let SO(n) be the group of real orthogonal matrices of determinant one. Define [Delta](A)={det(AoQ):Q[set membership, variant]SO(n)}, where o denotes the Hadamard product of matrices. For a permutation [sigma] on {1,...,n}, define It is shown that if the equation z[sigma]=det(AoQ) has in SO(n) only the obvious solutions (Q=([epsilon]i[delta][sigma]i,j), [epsilon]i=±1 such that [epsilon]1...[epsilon]n=sgn[sigma]), then the local shape of [Delta](A) in a vicinity of z[sigma] resembles a truncated cone whose opening angle equals , where [sigma]1, [sigma]2 differ from [sigma] by transpositions. This lends further credibility to the well known de Oliveira Marcus Conjecture (OMC) concerning the determinant of the sum of normal n×n matrices. We deduce the mentioned fact from a general result concerning multivariate power series and also use some elementary algebraic topology.http://www.sciencedirect.com/science/article/B6V0R-4NJG44V-3/1/29cc71d6352bcfea422c3dc7beebcbc

The inequality of Milne and its converse II

We prove the following let α,β,a > 0, and b < 0 be real numbers, and let Wj (j = 1,...,n; n ≥2) be positive real numbers with w1+ ⋯+wn= 1. The inequalities α ∑j=1n wj/(1- pja) ≤ ∑j=1n wj/(1 - pj) ∑ j=1n wj/(1+pj) ≤ β ∑j=1n wj/(1-pjb) hold for all real numbers pj ∈ [0,1) (j = 1,...,n) if and only if α ≤ min(1,a/2) and β ≥ max(1,(1 -min 1≤j≤nwj/2)b). Furthermore, we provide a matrix version. The first inequality (with α = 1 and a = 2) is a discrete counterpart of an integral inequality published by E. A. Milne in 1925